If $A : B = \frac{1}{2} : \frac{3}{8}$, $B : C = \frac{1}{3} : \frac{5}{9}$ and $C : D = \frac{5}{6} : \frac{3}{4}$, then the ratio $A : B : C : D$ is

### Concept & Simplifying Fractional Ratios To combine ratios given as fractions, first simplify each ratio into whole numbers by multiplying both parts by the least common multiple (LCM) of their denominators. $$ \frac{x/y}{a/b} = \frac{xb}{ya} $$ ### Step-by-Step Solution 1. Simplify $A : B = \frac{1}{2} : \frac{3}{8}$. The LCM of 2 and 8 is 8. Multiply both fractions by 8: $A : B = 4 : 3$ 2. Simplify $B : C = \frac{1}{3} : \frac{5}{9}$. The LCM of 3 and 9 is 9. Multiply both fractions by 9: $B : C = 3 : 5$ 3. Simplify $C : D = \frac{5}{6} : \frac{3}{4}$. The LCM of 6 and 4 is 12. Multiply both fractions by 12: $C : D = 10 : 9$ 4. Now, combine the standard ratios $A : B = 4 : 3$ and $B : C = 3 : 5$. The common term is $B$ (values 3 and 3). Since they already match, we can directly link them: $A : B : C = 4 : 3 : 5$ 5. Combine $A : B : C = 4 : 3 : 5$ with $C : D = 10 : 9$. The common term is $C$ (values 5 and 10). The LCM is 10. 6. Scale the first ratio by multiplying it by 2 to make $C$ equal to 10: $A : B : C = 8 : 6 : 10$ 7. Link it with $C : D$ since $C$ now matches: $A : B : C : D = 8 : 6 : 10 : 9$ ### Exam Strategy & Shortcut Simplify the fractions instantly in your head. Then write them in a cascading structure. Since $B$ naturally aligns as $3$ in both initial ratios, $A:B:C$ trivially becomes $4:3:5$. You then only need to match $C=5$ with $C=10$ by doubling the first set, producing $8:6:10$ and appending the $9$. ### Common Pitfall Attempting to equalize the intermediate terms while the ratios are still in their original fractional forms is a massive recipe for calculation errors and lost time. Always convert to integer ratios immediately. ### Final Answer Therefore, the correct answer is **$8 : 6 : 10 : 9$**.